AP Calculus AB Flashcards: 8-Unit Formulas, Theorems & Concepts

Bu destedeki kartlar

1. Kart 1

   Soru

   What does $\lim_{x \to a} f(x)=L$ say?

   Cevap

   The values of $f(x)$ approach $L$ as $x$ approaches $a$ from both sides. The statement doesn't require $f(a)=L$ or even require $f(a)$ to exist.

2. Kart 2

   Soru

   How can a table estimate $\lim_{x \to a} f(x)$?

   Cevap

   Use inputs approaching $a$ from below and above, then look for a common output value. Values exactly at $x=a$ don't determine the limit.

3. Kart 3

   Soru

   When does direct substitution evaluate a limit?

   Cevap

   When the function is continuous at the target input. Then

   $$\lim_{x \to a} f(x)=f(a).$$

4. Kart 4

   Soru

   Three conditions for continuity at $x=a$?

   Cevap

   $f(a)$ exists, $\lim_{x \to a}f(x)$ exists, and

   $$\lim_{x \to a}f(x)=f(a).$$

5. Kart 5

   Soru

   Intermediate Value Theorem: hypotheses and conclusion?

   Cevap

   If $f$ is continuous on $[a,b]$ and $N$ lies between $f(a)$ and $f(b)$, then some $c$ in $[a,b]$ satisfies $f(c)=N$. If $N$ is strictly between the endpoint values, $c$ lies in $(a,b)$.

6. Kart 6

   Soru

   When does a two-sided limit equal $L$?

   Cevap

   Exactly when both one-sided limits equal $L$:

   $$\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L.$$

   If the one-sided limits differ, the two-sided limit doesn't exist.

7. Kart 7

   Soru

   How do you read a finite limit from a graph?

   Cevap

   Follow the graph toward the target $x$-value from both sides. The common approached $y$-value is the limit, regardless of a hole or a differently placed filled point.

8. Kart 8

   Soru

   Limit law for a sum or difference?

   Cevap

   If both component limits exist,

   $$\lim_{x\to a}[f(x)\pm g(x)]=\lim_{x\to a}f(x)\pm\lim_{x\to a}g(x).$$

9. Kart 9

   Soru

   What makes a discontinuity removable?

   Cevap

   The finite two-sided limit exists, but the function value is missing or differs from that limit. Redefining the function at one point can make it continuous.

10. Kart 10

    Soru

    Squeeze Theorem: usable form?

    Cevap

    If $g(x)\le f(x)\le h(x)$ near $a$ and

    $$\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L,$$

    then $\lim_{x\to a}f(x)=L$.

11. Kart 11

    Soru

    What does $\lim_{x\to a}f(x)=+\infty$ mean?

    Cevap

    $f(x)$ grows without bound above as $x$ approaches $a$. It describes unbounded behavior, not a finite limit value.

12. Kart 12

    Soru

    What must a table show for a left-hand limit?

    Cevap

    Inputs less than the target and moving toward it. For $\lim_{x\to a^-}f(x)$, use $x<a$ with $x$ getting closer to $a$.

13. Kart 13

    Soru

    Limit law for a product?

    Cevap

    If both limits exist,

    $$\lim_{x\to a}[f(x)g(x)]=\left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a}g(x)\right).$$

14. Kart 14

    Soru

    Graph signature of a jump discontinuity?

    Cevap

    The left- and right-hand limits are finite but unequal. The function approaches different heights from the two sides.

15. Kart 15

    Soru

    Which theorem can guarantee a root on $[a,b]$?

    Cevap

    The Intermediate Value Theorem. If $f$ is continuous on $[a,b]$ and $0$ lies between $f(a)$ and $f(b)$, then $f(c)=0$ for some $c$ in the interval.

16. Kart 16

    Soru

    Horizontal asymptote from a limit at infinity?

    Cevap

    If $\lim_{x\to\infty}f(x)=L$ or $\lim_{x\to-\infty}f(x)=L$, then $y=L$ is a horizontal asymptote in that direction.

17. Kart 17

    Soru

    What does an open circle say about a graph's limit?

    Cevap

    Only that the displayed point isn't included there. The limit depends on nearby behavior; it may still exist and equal the hole's height.

18. Kart 18

    Soru

    Limit law for a quotient—and its condition?

    Cevap

    If both limits exist and the denominator limit is nonzero,

    $$\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}.$$

    The law doesn't apply when the denominator limit is $0$.

19. Kart 19

    Soru

    What does continuity on $[a,b]$ require at the endpoints?

    Cevap

    Continuity on $(a,b)$, right-continuity at $a$, and left-continuity at $b$:

    $$\lim_{x\to a^+}f(x)=f(a),\qquad \lim_{x\to b^-}f(x)=f(b).$$

20. Kart 20

    Soru

    When is the Squeeze Theorem a natural choice?

    Cevap

    When a hard or oscillating expression can be trapped between two simpler expressions with the same limit.

21. Kart 21

    Soru

    Vertical asymptote from one-sided behavior?

    Cevap

    If at least one one-sided limit at $x=a$ is $+\infty$ or $-\infty$, then $x=a$ is a vertical asymptote.

22. Kart 22

    Soru

    Limit at infinity of equal-degree rational functions?

    Cevap

    The ratio of the leading coefficients:

    $$\lim_{x\to\pm\infty}\frac{a_nx^n+\cdots}{b_nx^n+\cdots}=\frac{a_n}{b_n}.$$

    This assumes $b_n\ne0$.

23. Kart 23

    Soru

    When can a limit pass through a continuous outer function?

    Cevap

    If $\lim_{x\to a}g(x)=L$ and $f$ is continuous at $L$, then

    $$\lim_{x\to a}f(g(x))=f(L).$$

24. Kart 24

    Soru

    What makes a discontinuity infinite?

    Cevap

    The function becomes unbounded on at least one side of the input, usually producing a vertical asymptote.

25. Kart 25

    Soru

    Left limit $=2$ and right limit $=5$: two-sided limit?

    Cevap

    It doesn't exist. A finite two-sided limit requires the two one-sided limits to agree.

26. Kart 26

    Soru

    Standard trigonometric limit behind $\frac{\sin x}{x}$?

    Cevap

    With angles in radians,

    $$\lim_{x\to0}\frac{\sin x}{x}=1.$$

    Equivalent scaled forms follow by substitution.

27. Kart 27

    Soru

    Continuity of a composition?

    Cevap

    If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f\circ g$ is continuous at $a$.

28. Kart 28

    Soru

    Limit at infinity when a rational numerator has lower degree?

    Cevap

    $0$. If the numerator's degree is less than the denominator's, the denominator dominates as $x\to\pm\infty$.

29. Kart 29

    Soru

    What does the indeterminate form $0/0$ tell you?

    Cevap

    Direct substitution hasn't determined the limit. Simplify, use an appropriate limit method, or inspect another representation; $0/0$ isn't the limit value.

30. Kart 30

    Soru

    When do opposite infinite one-sided limits give a two-sided limit?

    Cevap

    They don't. For example, $+\infty$ from the left and $-\infty$ from the right mean the two-sided limit doesn't exist, though a vertical asymptote may be present.

31. Kart 31

    Soru

    How do you choose a parameter to make a piecewise function continuous?

    Cevap

    Set the relevant one-sided formulas and the function value equal at the join, then solve the resulting equation for the parameter.

32. Kart 32

    Soru

    Value of $\lim_{x\to0}\frac{1-\cos x}{x}$?

    Cevap

    $0$. Rationalizing gives a product involving $\sin x/x$ and a factor that approaches $0$.

33. Kart 33

    Soru

    Can $\lim_{x\to a}f(x)$ exist when $f(a)$ doesn't?

    Cevap

    Yes. A limit uses nearby values, so a hole at $x=a$ can coexist with a finite two-sided limit.

34. Kart 34

    Soru

    What graph behavior makes a finite limit fail even without a jump?

    Cevap

    Unbounded or persistent oscillating behavior near the input can prevent a finite limit. Check both sides rather than relying on the plotted point.

35. Kart 35

    Soru

    Average rate of change of $f$ on $[a,b]$?

    Cevap

    $$\frac{f(b)-f(a)}{b-a}$$

    It is the slope of the secant line through $(a,f(a))$ and $(b,f(b))$.

36. Kart 36

    Soru

    Derivative at $x=a$ using an increment $h$?

    Cevap

    $$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$

    The derivative exists only if this finite limit exists.

37. Kart 37

    Soru

    Tangent-line equation to $y=f(x)$ at $x=a$?

    Cevap

    $$y-f(a)=f'(a)(x-a)$$

    This requires $f'(a)$ to exist.

38. Kart 38

    Soru

    What does differentiability imply about continuity?

    Cevap

    If $f$ is differentiable at $a$, then $f$ is continuous at $a$. The converse is false: continuity alone doesn't guarantee differentiability.

39. Kart 39

    Soru

    Power rule for derivatives?

    Cevap

    $$\frac{d}{dx}x^n=nx^{n-1}$$

    Apply it where the original real-valued power function and its derivative are defined.

40. Kart 40

    Soru

    Units of $f'(x)$?

    Cevap

    Output units of $f$ per input unit of $x$. A derivative is a rate of change, so its units are a quotient.

41. Kart 41

    Soru

    Derivative at $x=a$ using $x\to a$?

    Cevap

    $$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$

    This is equivalent to the $h$-form after setting $h=x-a$.

42. Kart 42

    Soru

    How does a graph of $f$ show the sign of $f'$?

    Cevap

    $f'(x)>0$ where $f$ rises as $x$ increases, and $f'(x)<0$ where $f$ falls. A horizontal tangent gives $f'(x)=0$ when the derivative exists.

43. Kart 43

    Soru

    Derivative of a constant?

    Cevap

    $$\frac{d}{dx}C=0$$

    A constant function has zero rate of change.

44. Kart 44

    Soru

    Derivative of $\sin x$?

    Cevap

    $$\frac{d}{dx}(\sin x)=\cos x$$

    The angle must be measured in radians for the standard formula.

45. Kart 45

    Soru

    Product rule?

    Cevap

    $$\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$$

    Differentiating each factor and multiplying the results is not the product rule.

46. Kart 46

    Soru

    How can nearby table values estimate $f'(a)$?

    Cevap

    Use a difference quotient with inputs close to $a$. A symmetric estimate is

    $$f'(a)\approx\frac{f(a+h)-f(a-h)}{2h}.$$

    Smaller $h$ often helps, subject to the data's precision.

47. Kart 47

    Soru

    What does $f''(x)$ measure?

    Cevap

    The rate of change of $f'(x)$ with respect to $x$. Its units are the units of $f$ per square input unit.

48. Kart 48

    Soru

    Instantaneous rate of change of $f$ at $a$?

    Cevap

    $f'(a)$. It is the limit of average rates over intervals shrinking to $a$, and geometrically it is the tangent-line slope.

49. Kart 49

    Soru

    Derivative of a sum or difference?

    Cevap

    $$\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x)$$

50. Kart 50

    Soru

    Derivative of $\cos x$?

    Cevap

    $$\frac{d}{dx}(\cos x)=-\sin x$$

    The standard formula assumes radians.

51. Kart 51

    Soru

    Quotient rule?

    Cevap

    For $g(x)\ne0$,

    $$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}.$$

    The order in the numerator matters.

52. Kart 52

    Soru

    Common notations for the first derivative?

    Cevap

    $f'(x)$, $y'$, $\dfrac{dy}{dx}$, and $\dfrac{d}{dx}f(x)$. They describe the same derivative in different contexts.

53. Kart 53

    Soru

    Derivative of $e^x$?

    Cevap

    $$\frac{d}{dx}e^x=e^x$$

54. Kart 54

    Soru

    What graph features can make $f$ nondifferentiable?

    Cevap

    A discontinuity, corner, cusp, vertical tangent, or other failure of the difference-quotient limit. Continuity is necessary but not sufficient.

55. Kart 55

    Soru

    Derivative of $\tan x$?

    Cevap

    Where $\tan x$ is defined,

    $$\frac{d}{dx}(\tan x)=\sec^2x.$$

    Angles are in radians.

56. Kart 56

    Soru

    What does the derivative function $f'$ assign to each input?

    Cevap

    The instantaneous rate of change—or tangent slope—of $f$ at that input, wherever the derivative exists.

57. Kart 57

    Soru

    Derivative of $\ln x$?

    Cevap

    For $x>0$,

    $$\frac{d}{dx}\ln x=\frac1x.$$

    More generally, $d(\ln|x|)/dx=1/x$ for $x\ne0$.

58. Kart 58

    Soru

    How does the power rule handle roots or negative powers?

    Cevap

    Rewrite them as $x^n$ and apply $nx^{n-1}$ on intervals where the real-valued expression is defined. Domain restrictions still matter.

59. Kart 59

    Soru

    Derivative of $\csc x$?

    Cevap

    Where $\csc x$ is defined,

    $$\frac{d}{dx}(\csc x)=-\csc x\cot x.$$

    Angles are in radians.

60. Kart 60

    Soru

    If $f'(x)>0$ throughout an interval, what does $f$ do there?

    Cevap

    $f$ is increasing on that interval.

61. Kart 61

    Soru

    Derivative of $a^x$ for a constant base?

    Cevap

    For $a>0$,

    $$\frac{d}{dx}a^x=a^x\ln a.$$

    When $a=1$, the derivative is $0$.

62. Kart 62

    Soru

    How can a graph estimate $f'(a)$?

    Cevap

    Estimate the slope of the tangent line at $x=a$, using two convenient points on a drawn tangent when available. A steeper tangent has a derivative with larger magnitude.

63. Kart 63

    Soru

    Derivative of $\sec x$?

    Cevap

    Where $\sec x$ is defined,

    $$\frac{d}{dx}(\sec x)=\sec x\tan x.$$

    Angles are in radians.

64. Kart 64

    Soru

    If $f''(x)>0$, how is $f'$ changing?

    Cevap

    $f'$ is increasing. This is also the derivative condition associated with $f$ being concave up.

65. Kart 65

    Soru

    Derivative of $\log_a x$?

    Cevap

    For $x>0$, $a>0$, and $a\ne1$,

    $$\frac{d}{dx}\log_a x=\frac{1}{x\ln a}.$$

66. Kart 66

    Soru

    Product rule from a table at $x=a$?

    Cevap

    For $h=fg$,

    $$h'(a)=f'(a)g(a)+f(a)g'(a).$$

    Use the four table entries at the same input.

67. Kart 67

    Soru

    Derivative of $\cot x$?

    Cevap

    Where $\cot x$ is defined,

    $$\frac{d}{dx}(\cot x)=-\csc^2x.$$

    Angles are in radians.

68. Kart 68

    Soru

    Why isn't $|x|$ differentiable at $x=0$?

    Cevap

    Its left-hand slope is $-1$ and right-hand slope is $1$. The one-sided derivative limits disagree, creating a corner.

69. Kart 69

    Soru

    Constant-multiple rule?

    Cevap

    For a constant $c$,

    $$\frac{d}{dx}[c f(x)]=c f'(x).$$

70. Kart 70

    Soru

    Quotient rule from a table at $x=a$?

    Cevap

    For $h=f/g$ with $g(a)\ne0$,

    $$h'(a)=\frac{f'(a)g(a)-f(a)g'(a)}{[g(a)]^2}.$$

71. Kart 71

    Soru

    Chain rule for $f(g(x))$?

    Cevap

    $$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$$

    Differentiate the outer function at the unchanged inner input, then multiply by the inner derivative.

72. Kart 72

    Soru

    How do you identify inner and outer functions in a composite?

    Cevap

    Find the expression evaluated first—that is the inner function. The operation applied to its result is the outer function.

73. Kart 73

    Soru

    Core rule when differentiating an implicit equation in $x$ and $y$?

    Cevap

    Treat $y$ as a differentiable function of $x$. Every derivative of an expression involving $y$ gains a factor of $dy/dx$ by the chain rule.

74. Kart 74

    Soru

    Derivative of an inverse function at $x$?

    Cevap

    If $f$ is differentiable and one-to-one near $f^{-1}(x)$, with $f'(f^{-1}(x))\ne0$,

    $$[f^{-1}]'(x)=\frac{1}{f'(f^{-1}(x))}.$$

75. Kart 75

    Soru

    Derivative of $\arcsin x$?

    Cevap

    For $|x|<1$,

    $$\frac{d}{dx}(\arcsin x)=\frac{1}{\sqrt{1-x^2}}.$$

76. Kart 76

    Soru

    Notation for the third derivative of $f$?

    Cevap

    $f'''(x)$ or $\dfrac{d^3f}{dx^3}$. The exponent on $d$ indicates derivative order; it is not an ordinary power.

77. Kart 77

    Soru

    If $h(x)=f(g(x))$, what table entries give $h'(a)$?

    Cevap

    $$h'(a)=f'(g(a))g'(a)$$

    Use $g(a)$ to find the input needed for the table entry of $f'$.

78. Kart 78

    Soru

    For $x^2+y^2=r^2$, what is $dy/dx$?

    Cevap

    Where $y\ne0$,

    $$\frac{dy}{dx}=-\frac{x}{y}.$$

    Differentiate to get $2x+2y(dy/dx)=0$.

79. Kart 79

    Soru

    If $f(a)=b$, how do you find $[f^{-1}]'(b)$?

    Cevap

    Provided $f'(a)\ne0$,

    $$[f^{-1}]'(b)=\frac{1}{f'(a)}.$$

    The inverse swaps the input-output pair $(a,b)$.

80. Kart 80

    Soru

    Derivative of $\arctan x$?

    Cevap

    For every real $x$,

    $$\frac{d}{dx}(\arctan x)=\frac{1}{1+x^2}.$$

81. Kart 81

    Soru

    Derivative of $e^{g(x)}$?

    Cevap

    $$\frac{d}{dx}e^{g(x)}=e^{g(x)}g'(x)$$

    The extra factor is the chain rule.

82. Kart 82

    Soru

    Slope of a tangent to an implicit curve $F(x,y)=0$?

    Cevap

    Differentiate the relation with respect to $x$, solve for $dy/dx$, then substitute the point. Confirm the point lies on the curve and the resulting slope is defined.

83. Kart 83

    Soru

    Why must $f'(a)\ne0$ to use $[f^{-1}]'(f(a))=1/f'(a)$?

    Cevap

    Because the reciprocal slope would be undefined when $f'(a)=0$. The inverse may have a vertical tangent or fail to be differentiable there.

84. Kart 84

    Soru

    Derivative of $\arccos x$?

    Cevap

    For $|x|<1$,

    $$\frac{d}{dx}(\arccos x)=-\frac{1}{\sqrt{1-x^2}}.$$

85. Kart 85

    Soru

    How do you find $d^2y/dx^2$ for an implicit relation?

    Cevap

    Differentiate the first-derivative equation again with respect to $x$, include $dy/dx$ factors, then substitute the known expression for $dy/dx$ if needed.

86. Kart 86

    Soru

    Derivative of $\ln(g(x))$?

    Cevap

    Where $g(x)>0$,

    $$\frac{d}{dx}\ln(g(x))=\frac{g'(x)}{g(x)}.$$

    For $\ln|g(x)|$, the same derivative holds where $g(x)\ne0$.

87. Kart 87

    Soru

    Derivative of $y^n$ when $y=y(x)$?

    Cevap

    $$\frac{d}{dx}(y^n)=ny^{n-1}\frac{dy}{dx}$$

    The $dy/dx$ factor comes from the chain rule.

88. Kart 88

    Soru

    How are tangent slopes of inverse graphs related?

    Cevap

    At reflected points $(a,b)$ and $(b,a)$, the slopes are reciprocals when both are defined and nonzero.

89. Kart 89

    Soru

    Derivative of $\arcsin(g(x))$?

    Cevap

    $$\frac{d}{dx}\arcsin(g(x))=\frac{g'(x)}{\sqrt{1-[g(x)]^2}},\qquad |g(x)|<1.$$

90. Kart 90

    Soru

    Derivative of $\sin(g(x))$?

    Cevap

    $$\frac{d}{dx}\sin(g(x))=\cos(g(x))g'(x)$$

91. Kart 91

    Soru

    Horizontal tangent on an implicit curve: derivative condition?

    Cevap

    $dy/dx=0$ at a valid point, with the derivative defined there. In a fraction for $dy/dx$, the numerator is typically zero while the denominator is nonzero.

92. Kart 92

    Soru

    How do you differentiate $f^{-1}(x)$ without solving for the inverse?

    Cevap

    Use the reciprocal derivative formula and the matching original input: find $a$ with $f(a)=x$, then compute $1/f'(a)$.

93. Kart 93

    Soru

    Difference between $f''(x)$ and $[f'(x)]^2$?

    Cevap

    $f''(x)$ is the derivative of $f'(x)$. The expression $[f'(x)]^2$ is the square of the first derivative; they are generally unrelated.

94. Kart 94

    Soru

    Derivative of $[g(x)]^n$?

    Cevap

    $$\frac{d}{dx}[g(x)]^n=n[g(x)]^{n-1}g'(x)$$

    This combines the power rule with the chain rule.

95. Kart 95

    Soru

    Vertical tangent on an implicit curve: derivative clue?

    Cevap

    $dy/dx$ becomes unbounded or undefined while the curve still has a vertical tangent. In a simplified derivative fraction, the denominator is often zero and the numerator nonzero.

96. Kart 96

    Soru

    Table formula for an inverse derivative at $x=b$?

    Cevap

    Find $a$ in the table with $f(a)=b$. If $f'(a)\ne0$, then

    $$[f^{-1}]'(b)=\frac{1}{f'(a)}.$$

97. Kart 97

    Soru

    Derivative of $\arctan(g(x))$?

    Cevap

    $$\frac{d}{dx}\arctan(g(x))=\frac{g'(x)}{1+[g(x)]^2}$$

98. Kart 98

    Soru

    How do product and chain rules combine in $f(x)g(h(x))$?

    Cevap

    $$\frac{d}{dx}[f(x)g(h(x))]=f'(x)g(h(x))+f(x)g'(h(x))h'(x).$$

    Use the product rule outside and the chain rule on the composite factor.

99. Kart 99

    Soru

    Why can $d^2y/dx^2$ depend on both $x$ and $y$?

    Cevap

    An implicit relation may never solve explicitly for $y$. After differentiating twice and replacing $dy/dx$, the result can naturally remain a function of both coordinates.

100. Kart 100

     Soru

     A quantity $y$ changes through $u$, which changes with $x$. How are the rates connected?

     Cevap

     When the functions are differentiable, the chain rule gives

     $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.$$

101. Kart 101

     Soru

     What local property lets a function have an inverse derivative?

     Cevap

     The function must be one-to-one near the point, differentiable there, and have a nonzero derivative. Then the inverse is locally differentiable with reciprocal slope.

102. Kart 102

     Soru

     Derivative of $a^{g(x)}$?

     Cevap

     For $a>0$,

     $$\frac{d}{dx}a^{g(x)}=a^{g(x)}\ln(a)\,g'(x).$$

103. Kart 103

     Soru

     How should $Q'(t)$ be interpreted in context?

     Cevap

     At time $t$, the quantity $Q$ changes at an instantaneous rate of $Q'(t)$ output units per unit of time. Include the quantity, time, direction or sign, and units.

104. Kart 104

     Soru

     Position, velocity, and acceleration relationships?

     Cevap

     For position $s(t)$,

     $$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t).$$

105. Kart 105

     Soru

     Central idea of a related-rates problem?

     Cevap

     Write an equation connecting the changing quantities, differentiate it with respect to time, then use the data for the specified instant.

106. Kart 106

     Soru

     Linearization of $f$ near $x=a$?

     Cevap

     $$L(x)=f(a)+f'(a)(x-a)$$

     For $x$ close to $a$, $f(x)\approx L(x)$.

107. Kart 107

     Soru

     L’Hospital’s Rule: basic conditions?

     Cevap

     For a quotient with differentiable numerator and denominator near the target, denominator derivative nonzero nearby, and indeterminate form $0/0$ or $\infty/\infty$, the original limit equals the limit of the derivative quotient when that latter limit exists or is infinite.

108. Kart 108

     Soru

     If distance is in meters and time in seconds, units of acceleration?

     Cevap

     Meters per second squared, $\text{m}/\text{s}^2$. Acceleration is the rate of change of velocity with respect to time.

109. Kart 109

     Soru

     Speed in terms of velocity?

     Cevap

     $$\text{speed}=|v(t)|$$

     Velocity includes direction; speed is nonnegative magnitude.

110. Kart 110

     Soru

     Why do $x$ and $y$ gain $dx/dt$ and $dy/dt$ in related rates?

     Cevap

     They are functions of time. Differentiating an expression such as $x^2$ with respect to $t$ gives $2x(dx/dt)$ by the chain rule.

111. Kart 111

     Soru

     Differential approximation connecting $dx$ and $dy$?

     Cevap

     $$dy=f'(x)\,dx$$

     For a small change $\Delta x$, the actual change satisfies $\Delta y\approx f'(x)\Delta x$.

112. Kart 112

     Soru

     Which indeterminate forms directly allow L’Hospital’s Rule?

     Cevap

     $0/0$ and $\infty/\infty$. Other indeterminate forms must first be rewritten as an appropriate quotient.

113. Kart 113

     Soru

     How do you estimate an instantaneous contextual rate from a table?

     Cevap

     Use a nearby difference quotient, preferably with times on both sides of the target. Report the result with output units per time unit.

114. Kart 114

     Soru

     What does positive acceleration say about velocity?

     Cevap

     Velocity is increasing. It does not by itself say the object moves forward or speeds up; the sign of velocity also matters.

115. Kart 115

     Soru

     Related rates: when should numerical values be substituted?

     Cevap

     After differentiating the general relationship with respect to time. Substituting fixed values too early can erase the rates being sought.

116. Kart 116

     Soru

     How does concavity predict linearization error?

     Cevap

     Near the tangency point, a concave-up graph generally lies above its tangent line, so the linearization underestimates. A concave-down graph generally lies below, so it overestimates.

117. Kart 117

     Soru

     Why can't L’Hospital’s Rule be applied directly to a product?

     Cevap

     The rule applies to quotients with $0/0$ or $\infty/\infty$ form. Rewrite an indeterminate product such as $0\cdot\infty$ as a quotient first.

118. Kart 118

     Soru

     When is a particle moving in the positive direction?

     Cevap

     When $v(t)>0$. Position then increases as time increases.

119. Kart 119

     Soru

     How can velocity show a change of direction?

     Cevap

     Velocity changes sign. A time with $v(t)=0$ is only a candidate; confirm the sign differs on the two sides.

120. Kart 120

     Soru

     First equation to seek in a geometric related-rates problem?

     Cevap

     A geometric constraint involving the changing quantities, such as a Pythagorean, area, volume, or similar-triangle relation.

121. Kart 121

     Soru

     Tangent-line approximation of $f(a+\Delta x)$?

     Cevap

     $$f(a+\Delta x)\approx f(a)+f'(a)\Delta x$$

     It is most reliable for small $|\Delta x|$ where the function is well approximated by its tangent.

122. Kart 122

     Soru

     When may L’Hospital’s Rule be applied more than once?

     Cevap

     When the derivative quotient still has $0/0$ or $\infty/\infty$ form and the rule's conditions continue to hold.

123. Kart 123

     Soru

     What must a contextual derivative sentence include?

     Cevap

     The changing quantity, the instant or input, the rate's sign or direction when relevant, and correct output-per-input units.

124. Kart 124

     Soru

     Velocity negative and acceleration positive: what happens?

     Cevap

     The particle moves in the negative direction while its velocity increases toward zero. Its speed decreases as long as velocity and acceleration have opposite signs.

125. Kart 125

     Soru

     How should a negative related rate be interpreted?

     Cevap

     The measured quantity is decreasing at that instant. Keep the sign and state the units; don't silently report only its magnitude.

126. Kart 126

     Soru

     When is local linearity a sound approximation tool?

     Cevap

     When $f$ is differentiable near the base point and the target input is close enough that curvature has limited effect.

127. Kart 127

     Soru

     Can L’Hospital’s Rule handle a one-sided limit?

     Cevap

     Yes, if the required indeterminate form and differentiability conditions hold on that side and the derivative-quotient limit exists or is infinite.

128. Kart 128

     Soru

     When is speed increasing?

     Cevap

     When velocity and acceleration have the same sign, so $v(t)a(t)>0$.

129. Kart 129

     Soru

     Volume changes with time: notation for its rate?

     Cevap

     $dV/dt$. Its units are cubic length units per time unit.

130. Kart 130

     Soru

     Why are similar triangles useful in related rates?

     Cevap

     They can reduce several changing lengths to one constraint before differentiation, keeping the rate equation tied to the geometry.

131. Kart 131

     Soru

     Meaning of $dy=f'(x)dx$ in approximation?

     Cevap

     $dy$ is the tangent-line estimate of the actual output change $\Delta y$ caused by an input change $dx$.

132. Kart 132

     Soru

     What conclusion does L’Hospital’s Rule permit?

     Cevap

     Under its conditions,

     $$\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}.$$

     It does not say the two quotients are equal as functions.

133. Kart 133

     Soru

     When is speed decreasing?

     Cevap

     When velocity and acceleration have opposite signs, so $v(t)a(t)<0$.

134. Kart 134

     Soru

     What does a tangent slope read from a contextual graph represent?

     Cevap

     The instantaneous rate of the vertical quantity with respect to the horizontal quantity, with vertical units per horizontal unit.

135. Kart 135

     Soru

     Does $v(t)=0$ guarantee a particle changes direction?

     Cevap

     No. It is momentarily at rest, but direction changes only if velocity changes sign across that time.

136. Kart 136

     Soru

     How do you translate “$Q$ increases by 3 units per minute” into derivative notation?

     Cevap

     $dQ/dt=3$ in the stated time interval or at the stated instant. “Decreases by 3” would give $dQ/dt=-3$.

137. Kart 137

     Soru

     Extreme Value Theorem: hypothesis and conclusion?

     Cevap

     If $f$ is continuous on the closed interval $[a,b]$, then $f$ has at least one absolute minimum value and at least one absolute maximum value on $[a,b]$.

138. Kart 138

     Soru

     What is a critical number of $f$?

     Cevap

     A number $c$ in the domain of $f$ where $f'(c)=0$ or $f'(c)$ doesn't exist.

139. Kart 139

     Soru

     First derivative test for a local maximum?

     Cevap

     $f'$ changes from positive to negative at the critical point, so $f$ changes from increasing to decreasing.

140. Kart 140

     Soru

     Second-derivative sign for concave up?

     Cevap

     If $f''(x)>0$ on an interval, then $f$ is concave up there and $f'$ is increasing.

141. Kart 141

     Soru

     If the graph of $f'$ is above the $x$-axis, what does $f$ do?

     Cevap

     $f$ is increasing because $f'(x)>0$.

142. Kart 142

     Soru

     First step in an optimization model?

     Cevap

     Define the objective quantity and the constraint, including the feasible domain. Rewrite the objective as a function of one variable before differentiating.

143. Kart 143

     Soru

     Mean Value Theorem: hypotheses and conclusion?

     Cevap

     If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then some $c$ in $(a,b)$ satisfies

     $$f'(c)=\frac{f(b)-f(a)}{b-a}.$$

144. Kart 144

     Soru

     Candidates test for absolute extrema on $[a,b]$?

     Cevap

     Assuming $f$ is continuous on $[a,b]$, evaluate $f$ at every critical number in $(a,b)$ and at both endpoints. The largest value is the absolute maximum; the smallest is the absolute minimum.

     

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145. Kart 145

     Soru

     First derivative test for a local minimum?

     Cevap

     $f'$ changes from negative to positive at the critical point, so $f$ changes from decreasing to increasing.

146. Kart 146

     Soru

     What must happen at an inflection point?

     Cevap

     The graph's concavity changes. A zero or undefined value of $f''$ is only a candidate; verify a concavity change.

147. Kart 147

     Soru

     If $f'$ has a local maximum, what can that say about $f$?

     Cevap

     $f''$ may change from positive to negative there, so $f$ may change from concave up to concave down. Confirm the sign change rather than relying only on the point.

148. Kart 148

     Soru

     How do you confirm an optimization answer is absolute?

     Cevap

     Compare objective values at all feasible critical points and relevant endpoints, or use a justified monotonicity argument over the feasible domain.

149. Kart 149

     Soru

     Rolle’s Theorem: hypotheses and conclusion?

     Cevap

     If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then some $c$ in $(a,b)$ satisfies $f'(c)=0$.

150. Kart 150

     Soru

     Difference between absolute and relative extrema?

     Cevap

     An absolute maximum is at least every other value on the stated domain, and an absolute minimum is at most every other value. A relative extremum makes the corresponding comparison only with nearby values.

151. Kart 151

     Soru

     If $f$ is continuous at a critical number $c$ and $f'$ is positive on both sides, is there a local extremum?

     Cevap

     No. The function is increasing through $c$, so it has no local extremum there.

152. Kart 152

     Soru

     Second derivative test for a local minimum?

     Cevap

     If $f'(c)=0$ and $f''(c)>0$, then $f$ has a local minimum at $c$.

153. Kart 153

     Soru

     Zeros of $f'$ correspond to what features of $f$?

     Cevap

     Horizontal tangents where $f'$ exists. They are critical numbers and possible extrema, but each needs further sign or value analysis.

154. Kart 154

     Soru

     Implicit relation: how can $dy/dx$ reveal local behavior?

     Cevap

     Its sign shows whether the relation's local branch rises or falls as $x$ increases; zeros and undefined values mark possible horizontal or vertical tangents.

155. Kart 155

     Soru

     Which theorem links an average slope to an instantaneous slope?

     Cevap

     The Mean Value Theorem, provided the function is continuous on the closed interval and differentiable on its interior.

156. Kart 156

     Soru

     How can an implicit derivative locate a horizontal tangent?

     Cevap

     At a valid point on the relation, find where the simplified numerator of $dy/dx$ is zero while its denominator is nonzero, then confirm the local branch has the expected behavior.

157. Kart 157

     Soru

     Derivative-sign chart: where is $f$ decreasing?

     Cevap

     On intervals where $f'(x)<0$.

158. Kart 158

     Soru

     Second derivative test for a local maximum?

     Cevap

     If $f'(c)=0$ and $f''(c)<0$, then $f$ has a local maximum at $c$.

159. Kart 159

     Soru

     If $f'$ is increasing, what is the concavity of $f$?

     Cevap

     $f$ is concave up on that interval, assuming the relevant derivatives exist.

160. Kart 160

     Soru

     Why must an optimization domain be stated?

     Cevap

     The context can exclude algebraic candidates and add boundary cases. Absolute extrema depend on the feasible inputs, not just on where the derivative is zero.

161. Kart 161

     Soru

     Which theorem guarantees absolute extrema, not where they occur?

     Cevap

     The Extreme Value Theorem. Continuity on a closed interval guarantees at least one maximum value and one minimum value, but finding them requires analysis.

162. Kart 162

     Soru

     Can $f'(c)$ fail to exist at a local extremum?

     Cevap

     Yes. Corners, cusps, or other nondifferentiable domain points can be local extrema, which is why critical numbers include points where $f'$ is undefined.

163. Kart 163

     Soru

     For a function continuous at $c$, what same-sign pattern in $f'$ rules out a local extremum there?

     Cevap

     If $f'$ is positive on both sides of $c$, or negative on both sides, then $f$ keeps the same monotonic direction through $c$ and has no local extremum there.

164. Kart 164

     Soru

     If $f'(c)=0$ and $f''(c)=0$, what does the second derivative test conclude?

     Cevap

     Nothing. The test is inconclusive; use a first-derivative sign change, higher analysis, or direct comparison.

165. Kart 165

     Soru

     If the graph of $f'$ crosses from negative to positive, what feature does $f$ have?

     Cevap

     A local minimum at the crossing input, provided the input is in the domain of $f$.

166. Kart 166

     Soru

     How can an implicit derivative locate a vertical tangent?

     Cevap

     Find valid points where the simplified $dy/dx$ denominator is zero and numerator is nonzero, then confirm the curve has the expected local behavior.

167. Kart 167

     Soru

     Which extra condition turns the Mean Value Theorem into Rolle’s Theorem?

     Cevap

     $f(a)=f(b)$. The average slope becomes zero, so the guaranteed instantaneous slope is also zero.

168. Kart 168

     Soru

     Why are endpoints included in the candidates test?

     Cevap

     An absolute maximum or minimum on a closed interval can occur at an endpoint even though no two-sided derivative test applies there.

169. Kart 169

     Soru

     If $f'(x)=0$ throughout an interval, what is $f$ there?

     Cevap

     $f$ is constant on that interval, assuming the usual continuity and differentiability conditions that let the Mean Value Theorem apply.

170. Kart 170

     Soru

     Second-derivative sign for concave down?

     Cevap

     If $f''(x)<0$ on an interval, then $f$ is concave down there and $f'$ is decreasing.

171. Kart 171

     Soru

     Graph of $f'$ has a local minimum: possible effect on $f$?

     Cevap

     $f''$ may change from negative to positive, so $f$ may change from concave down to concave up. Verify the sign change.

172. Kart 172

     Soru

     What should the final line of an optimization solution state?

     Cevap

     The requested maximum or minimum quantity in context, with units and the feasible input that produces it when relevant.

173. Kart 173

     Soru

     Can Rolle’s Theorem be used if $f$ has a corner inside $(a,b)$?

     Cevap

     No. A corner breaks differentiability on the open interval, so the theorem's hypotheses are not satisfied.

174. Kart 174

     Soru

     How do $f''$ zeros help analyze a graph?

     Cevap

     They are candidates for changes in concavity. Test the sign of $f''$ on both sides; a zero alone doesn't guarantee an inflection point.

175. Kart 175

     Soru

     What does the accumulation function $F(x)=\int_a^x f(t)\,dt$ measure?

     Cevap

     The signed net accumulation of $f$ from $a$ to $x$. Contributions above the axis are positive; contributions below it are negative.

176. Kart 176

     Soru

     Left Riemann sum on equal subintervals?

     Cevap

     If $\Delta x=(b-a)/n$ and $x_i=a+i\Delta x$, then

     $$L_n=\sum_{i=1}^{n} f(x_{i-1})\Delta x.$$

177. Kart 177

     Soru

     What does $\int_a^b f(x)\,dx$ represent geometrically?

     Cevap

     Signed area between the graph and the $x$-axis from $a$ to $b$, when $f$ is integrable. Regions below the axis subtract from regions above it.

178. Kart 178

     Soru

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Cevap

     If $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$, then

     $$\int_a^b f(x)\,dx=F(b)-F(a).$$

179. Kart 179

     Soru

     Derivative of $F(x)=\int_a^x f(t)\,dt$?

     Cevap

     If $f$ is continuous, then

     $$F'(x)=f(x).$$

     This connects accumulation with instantaneous rate.

180. Kart 180

     Soru

     Why do all antiderivatives of the same function differ by a constant?

     Cevap

     If $F'=f$ and $G'=f$ on an interval, then $(F-G)'=0$, so $F-G=C$ on that interval.

181. Kart 181

     Soru

     Right Riemann sum on equal subintervals?

     Cevap

     If $\Delta x=(b-a)/n$ and $x_i=a+i\Delta x$, then

     $$R_n=\sum_{i=1}^{n} f(x_i)\Delta x.$$

182. Kart 182

     Soru

     How does reversing integral bounds change the value?

     Cevap

     It changes the sign:

     $$\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx.$$

183. Kart 183

     Soru

     Net Change Theorem?

     Cevap

     If $Q'(t)$ is the rate of change of a quantity, then

     $$Q(b)-Q(a)=\int_a^b Q'(t)\,dt.$$

184. Kart 184

     Soru

     Derivative of $\int_a^{g(x)} f(t)\,dt$?

     Cevap

     If $f$ is continuous on an interval containing $a$ and the range of $g$, and $g$ is differentiable, then

     $$\frac{d}{dx}\int_a^{g(x)} f(t)\,dt=f(g(x))g'(x).$$

185. Kart 185

     Soru

     Power rule for antiderivatives?

     Cevap

     For $n\ne-1$,

     $$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C.$$

186. Kart 186

     Soru

     Midpoint Riemann sum on equal subintervals?

     Cevap

     With midpoint $m_i=(x_{i-1}+x_i)/2$,

     $$M_n=\sum_{i=1}^{n} f(m_i)\Delta x.$$

187. Kart 187

     Soru

     How can an integral be split at an interior point $c$?

     Cevap

     For $a\le c\le b$,

     $$\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx.$$

188. Kart 188

     Soru

     Derivative of $G(x)=\int_x^b f(t)\,dt$?

     Cevap

     If $f$ is continuous, then

     $$G'(x)=-f(x).$$

     The variable lower bound produces the negative sign.

189. Kart 189

     Soru

     Antiderivative of $1/x$?

     Cevap

     On any interval not crossing zero,

     $$\int \frac1x\,dx=\ln|x|+C.$$

190. Kart 190

     Soru

     Trapezoidal approximation on equal subintervals?

     Cevap

     $$T_n=\frac{\Delta x}{2}\left[f(x_0)+2f(x_1)+\cdots+2f(x_{n-1})+f(x_n)\right].$$

191. Kart 191

     Soru

     How do geometric regions help evaluate a definite integral?

     Cevap

     Replace each familiar region with its geometric area, then attach a positive sign above the axis and a negative sign below it.

192. Kart 192

     Soru

     Basic antiderivatives of sine and cosine?

     Cevap

     $$\int \sin x\,dx=-\cos x+C,\qquad \int \cos x\,dx=\sin x+C.$$

193. Kart 193

     Soru

     Definite integral as a limit of Riemann sums?

     Cevap

     For an integrable function and sample points $x_i^*$,

     $$\int_a^b f(x)\,dx=\lim_{\max\Delta x_i\to0}\sum_i f(x_i^*)\Delta x_i.$$

194. Kart 194

     Soru

     Constant-multiple rule for integrals?

     Cevap

     For a constant $k$,

     $$\int_a^b kf(x)\,dx=k\int_a^b f(x)\,dx.$$

     The analogous rule holds for indefinite integrals.

195. Kart 195

     Soru

     What pattern suggests $u$-substitution?

     Cevap

     A composite expression paired with its derivative, such as $f(g(x))g'(x)$. Set $u=g(x)$ so $du=g'(x)\,dx$.

196. Kart 196

     Soru

     How should bounds change in a definite $u$-substitution?

     Cevap

     If $u=g(x)$, replace the $x$-bounds $a,b$ with $u$-bounds $g(a),g(b)$. Then finish entirely in $u$, or return to $x$ before applying the original bounds.

197. Kart 197

     Soru

     What condition makes $F(x)=\int_a^x f(t)\,dt$ differentiable with $F'(x)=f(x)$?

     Cevap

     Continuity of $f$ on an interval containing $a$ and $x$ is the standard AP Calculus condition.

198. Kart 198

     Soru

     Sum-and-difference rule for definite integrals?

     Cevap

     For integrable $f$ and $g$,

     $$\int_a^b [f(x)\pm g(x)]\,dx=\int_a^b f(x)\,dx\pm\int_a^b g(x)\,dx.$$

199. Kart 199

     Soru

     Basic antiderivative of $e^x$?

     Cevap

     $$\int e^x\,dx=e^x+C.$$

200. Kart 200

     Soru

     Basic antiderivatives of $\sec^2x$ and $\csc^2x$?

     Cevap

     $$\int\sec^2x\,dx=\tan x+C,\qquad \int\csc^2x\,dx=-\cot x+C.$$

201. Kart 201

     Soru

     For an increasing integrable function, how do left and right sums compare with the integral?

     Cevap

     On the same partition, the left sum is an underestimate and the right sum is an overestimate. Reverse those conclusions when the function is decreasing.

202. Kart 202

     Soru

     How does concavity predict trapezoidal and midpoint error?

     Cevap

     For a concave-up function, trapezoidal sums overestimate and midpoint sums underestimate. For a concave-down function, the directions reverse.

203. Kart 203

     Soru

     Why might polynomial long division help before integrating a rational function?

     Cevap

     When the numerator's degree is at least the denominator's, division rewrites the expression as a polynomial plus a simpler proper rational function.

204. Kart 204

     Soru

     What denominator pattern suggests an arctangent antiderivative?

     Cevap

     After completing the square and scaling, a form like

     $$\int\frac{1}{1+u^2}\,du=\arctan u+C.$$

205. Kart 205

     Soru

     Basic antiderivatives of $\sec x\tan x$ and $\csc x\cot x$?

     Cevap

     $$\int\sec x\tan x\,dx=\sec x+C,\qquad \int\csc x\cot x\,dx=-\csc x+C.$$

206. Kart 206

     Soru

     How does an initial condition determine an antiderivative?

     Cevap

     First find the family $F(x)+C$. Substitute the given point, such as $y(a)=b$, and solve for $C$.

207. Kart 207

     Soru

     Should a definite-integral answer include $+C$?

     Cevap

     No. A definite integral is one number or quantity. The constant of integration belongs to an indefinite integral or general antiderivative.

208. Kart 208

     Soru

     Why does an indefinite integral include $+C$?

     Cevap

     Differentiation loses additive constants. The $+C$ represents every function with the stated derivative.

209. Kart 209

     Soru

     When is $F(x)=\int_a^x f(t)\,dt$ increasing?

     Cevap

     Where $F'(x)=f(x)>0$. It is decreasing where $f(x)<0$.

210. Kart 210

     Soru

     How is the concavity of $F(x)=\int_a^x f(t)\,dt$ determined?

     Cevap

     Since $F'(x)=f(x)$, $F$ is concave up where $f$ is increasing and concave down where $f$ is decreasing, assuming the needed derivatives exist.

211. Kart 211

     Soru

     How is $\int_a^b |f(x)|\,dx$ interpreted?

     Cevap

     As total geometric area between $f$ and the $x$-axis. Split at zeros of $f$ and make every regional contribution nonnegative.

212. Kart 212

     Soru

     What constant-factor check completes many $u$-substitutions?

     Cevap

     Compare $du=g'(x)\,dx$ with the remaining factor. Multiply or divide by a constant so the integrand contains exactly the needed differential.

213. Kart 213

     Soru

     How do you recover $\Delta x$ from a sigma-form Riemann sum on $[a,b]$?

     Cevap

     Identify the factor multiplying each function value. For $n$ equal subintervals, it should be

     $$\Delta x=\frac{b-a}{n}.$$

214. Kart 214

     Soru

     Riemann sum for unequal subinterval widths?

     Cevap

     If $x_{i-1}$ to $x_i$ has width $\Delta x_i$ and sample point $x_i^*$, use

     $$\sum_i f(x_i^*)\Delta x_i.$$

215. Kart 215

     Soru

     Does continuity guarantee integrability on a closed interval?

     Cevap

     Yes. A function continuous on $[a,b]$ is integrable there. Some discontinuous functions are also integrable, so continuity is sufficient, not necessary.

216. Kart 216

     Soru

     Antiderivative pattern for $g'(x)/g(x)$?

     Cevap

     Where $g(x)\ne0$,

     $$\int\frac{g'(x)}{g(x)}\,dx=\ln|g(x)|+C.$$

217. Kart 217

     Soru

     What algebraic rewrites often reveal a basic antiderivative?

     Cevap

     Split sums, factor out constants, simplify rational powers, and rewrite radicals as exponents. Then apply known antiderivative rules term by term.

218. Kart 218

     Soru

     What units does $\int_a^b r(t)\,dt$ have?

     Cevap

     Rate units multiplied by input units. For example, liters per minute integrated over minutes gives liters.

219. Kart 219

     Soru

     What is a differential equation?

     Cevap

     An equation that relates an unknown function to one or more of its derivatives. A solution is a function that makes the equation true on an interval.

220. Kart 220

     Soru

     How does a verbal rate statement become a differential equation?

     Cevap

     Name the changing quantity and its independent variable, translate its rate as a derivative, and express that derivative using the stated relationship. For example, “rate proportional to $y$” becomes $dy/dt=ky$.

221. Kart 221

     Soru

     How do you verify that $y=f(x)$ solves a differential equation?

     Cevap

     Differentiate $f$ as needed, substitute $y$ and its derivatives into the equation, and confirm both sides agree on the claimed interval.

222. Kart 222

     Soru

     General solution versus particular solution?

     Cevap

     A general solution contains an arbitrary constant and represents a family of solution curves. A particular solution uses an initial condition to determine that constant.

223. Kart 223

     Soru

     What does one segment in a slope field show?

     Cevap

     At $(x,y)$, its slope equals the value of $dy/dx$ given by the differential equation at that point.

224. Kart 224

     Soru

     What units does the constant $k$ have in $dy/dt=ky$?

     Cevap

     Inverse time units, such as per hour. That makes the exponent $kt$ dimensionless.

225. Kart 225

     Soru

     How do you verify a proposed solution to an initial value problem?

     Cevap

     Check both requirements: the function must satisfy the differential equation throughout the stated interval, and it must satisfy the given initial condition.

226. Kart 226

     Soru

     What makes a first-order differential equation separable?

     Cevap

     It can be rearranged so all $y$ factors accompany $dy$ and all $x$ factors accompany $dx$, such as

     $$g(y)\,dy=f(x)\,dx.$$

227. Kart 227

     Soru

     What is an initial value problem?

     Cevap

     A differential equation paired with a value such as $y(x_0)=y_0$. It asks for a solution through $(x_0,y_0)$; depending on the equation, there may be zero, one, or multiple such solutions.

228. Kart 228

     Soru

     What is an isocline in a slope field?

     Cevap

     A curve along which the differential equation gives the same slope. For $dy/dx=F(x,y)$, an isocline satisfies $F(x,y)=k$ for a constant $k$.

229. Kart 229

     Soru

     How do you draw a slope-field segment at $(x_0,y_0)$?

     Cevap

     Evaluate the differential equation at $(x_0,y_0)$ to find the slope, then draw a short segment through that point with the resulting slope.

230. Kart 230

     Soru

     General solution of $dy/dt=ky$?

     Cevap

     $$y=Ce^{kt}$$

     for a constant $C$. The zero solution is included by $C=0$.

231. Kart 231

     Soru

     Core method for solving a separable differential equation?

     Cevap

     Separate the variables, integrate both sides, include a constant of integration, and solve for $y$ when practical. Then apply any initial condition.

232. Kart 232

     Soru

     How should a solution curve follow a slope field?

     Cevap

     It must pass through its initial point and remain tangent to the short field segments it crosses. It should not be drawn by connecting segment endpoints.

233. Kart 233

     Soru

     If $dy/dx=f(x)$, what pattern appears in its slope field?

     Cevap

     Every point with the same $x$-coordinate has the same slope, so the segments repeat vertically in columns.

234. Kart 234

     Soru

     Why is one integration constant enough after integrating both sides?

     Cevap

     Two constants can be combined: $C_2-C_1$ is still an arbitrary constant. Write a single $C$.

235. Kart 235

     Soru

     Solution of $dy/dt=ky$ with $y(0)=y_0$?

     Cevap

     $$y(t)=y_0e^{kt}.$$

236. Kart 236

     Soru

     Can one differential equation have infinitely many solutions?

     Cevap

     Yes. A differential equation commonly describes a family of solution functions. An initial condition may select one particular solution when the relevant conditions support uniqueness.

237. Kart 237

     Soru

     What is an equilibrium solution of $dy/dx=F(y)$?

     Cevap

     A constant solution $y=c$ where $F(c)=0$. In the slope field, the segments along that horizontal line have zero slope.

238. Kart 238

     Soru

     For continuous $f$, particular solution of $dy/dx=f(x)$ with $y(a)=b$?

     Cevap

     $$y(x)=b+\int_a^x f(t)\,dt.$$

     The Fundamental Theorem of Calculus gives $y'=f(x)$, and $y(a)=b$.

239. Kart 239

     Soru

     What can be lost when dividing to separate variables?

     Cevap

     Equilibrium solutions that make the divided factor zero. Check those constant solutions directly in the original differential equation.

240. Kart 240

     Soru

     In $dy/dt=ky$, what do the signs of $k$ mean?

     Cevap

     For a positive quantity, $k>0$ produces exponential growth, $k<0$ produces exponential decay, and $k=0$ keeps the quantity constant.

241. Kart 241

     Soru

     How can a table of slopes identify the matching differential equation?

     Cevap

     Test representative $(x,y)$ entries in each candidate equation. Match zeros, signs, and repeated slope patterns before checking exact numerical values.

242. Kart 242

     Soru

     How does the sign of $dy/dx$ describe a solution?

     Cevap

     The solution is increasing where $dy/dx>0$ and decreasing where $dy/dx<0$. At one point, $dy/dx=0$ gives a horizontal tangent; a constant $y=c$ is an equilibrium only when the derivative equation gives zero all along that level.

243. Kart 243

     Soru

     How can a differential equation determine a solution's concavity?

     Cevap

     Differentiate the equation with respect to the independent variable to obtain $y''$, using the chain rule for any $y$-dependence. Then use the sign of $y''$ along the solution.

244. Kart 244

     Soru

     Why must a differential-equation solution include an interval or domain?

     Cevap

     Algebraic steps may introduce restrictions, and the formula must remain differentiable and satisfy the original equation throughout the interval containing the initial point.

245. Kart 245

     Soru

     Doubling time for exponential growth $y=y_0e^{kt}$?

     Cevap

     For $k>0$,

     $$T_d=\frac{\ln2}{k}.$$

     It is independent of the initial amount.

246. Kart 246

     Soru

     How can a slope field reveal whether $dy/dx$ depends only on $y$?

     Cevap

     Slopes repeat horizontally: every point at the same height has the same segment slope.

247. Kart 247

     Soru

     How is an initial condition used after separation?

     Cevap

     Substitute the given $x$ and $y$ values into the integrated relationship to solve for the arbitrary constant, then state the resulting particular solution.

248. Kart 248

     Soru

     How do units check a model $dy/dt=F(t,y)$?

     Cevap

     The right side must have the same units as $dy/dt$: units of $y$ per unit of $t$. A mismatch signals an incorrect translation or parameter unit.

249. Kart 249

     Soru

     Why should a separated solution be checked in the original equation?

     Cevap

     Division, logarithms, square roots, or algebraic rearrangement can lose or introduce branches. Direct substitution confirms the formula and its stated domain.

250. Kart 250

     Soru

     Half-life for exponential decay $y=y_0e^{kt}$?

     Cevap

     For $k<0$,

     $$T_{1/2}=\frac{\ln(1/2)}{k}=\frac{\ln2}{|k|}.$$

251. Kart 251

     Soru

     Average value of $f$ on $[a,b]$?

     Cevap

     For integrable $f$ and $a<b$,

     $$f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx.$$

252. Kart 252

     Soru

     Displacement from velocity $v(t)$ on $[a,b]$?

     Cevap

     $$s(b)-s(a)=\int_a^b v(t)\,dt.$$

     Velocity below zero contributes negative displacement.

253. Kart 253

     Soru

     Area between vertical curves $y=f(x)$ and $y=g(x)$?

     Cevap

     On intervals where $f(x)\ge g(x)$,

     $$A=\int_a^b [f(x)-g(x)]\,dx.$$

     Think top minus bottom.

254. Kart 254

     Soru

     Volume from known cross-sectional area $A(x)$?

     Cevap

     If slices are perpendicular to the $x$-axis,

     $$V=\int_a^b A(x)\,dx.$$

255. Kart 255

     Soru

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Cevap

     If $f$ is continuous on $[a,b]$, then some $c\in[a,b]$ satisfies

     $$f(c)=\frac{1}{b-a}\int_a^b f(x)\,dx.$$

     If $a<b$, a point can also be chosen in $(a,b)$.

256. Kart 256

     Soru

     Velocity and acceleration from position $s(t)$?

     Cevap

     $$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t).$$

257. Kart 257

     Soru

     Cross-sectional area when each slice is a square?

     Cevap

     If the base segment has length $s(x)$, then

     $$A(x)=[s(x)]^2.$$

258. Kart 258

     Soru

     How do you find accumulation from an inflow rate and an outflow rate?

     Cevap

     Integrate the net rate:

     $$Q(b)-Q(a)=\int_a^b [r_{\text{in}}(t)-r_{\text{out}}(t)]\,dt.$$

259. Kart 259

     Soru

     Area between horizontal curves written as $x=R(y)$ and $x=L(y)$?

     Cevap

     On intervals where $R(y)\ge L(y)$,

     $$A=\int_c^d [R(y)-L(y)]\,dy.$$

     Think right minus left.

260. Kart 260

     Soru

     Disc-method volume formula?

     Cevap

     For radius $R(x)$ and slices perpendicular to the $x$-axis,

     $$V=\pi\int_a^b [R(x)]^2\,dx.$$

261. Kart 261

     Soru

     What units does average value have?

     Cevap

     The same units as the original function. Integration adds an input unit, and division by interval length removes it.

262. Kart 262

     Soru

     Total distance traveled from velocity $v(t)$?

     Cevap

     $$\text{distance}=\int_a^b |v(t)|\,dt.$$

     Split the interval wherever $v(t)=0$ and its sign changes.

263. Kart 263

     Soru

     Cross-sectional area when each slice is a rectangle?

     Cevap

     If the slice has base $b(x)$ and height $h(x)$, then

     $$A(x)=b(x)h(x).$$

     Use the stated relationship to express both in the integration variable.

264. Kart 264

     Soru

     How do you determine bounds for area between curves?

     Cevap

     Use the stated region boundaries or solve the curve-intersection equations. Check whether additional intersections inside the interval require splitting.

265. Kart 265

     Soru

     How is a rotation radius measured from a horizontal axis $y=k$?

     Cevap

     As vertical distance: $|y-k|$. For washers, identify which boundary stays farther from the axis over the interval.

266. Kart 266

     Soru

     When is a particle moving to the right or left?

     Cevap

     It moves right where $v(t)>0$ and left where $v(t)<0$. Position alone does not determine direction.

267. Kart 267

     Soru

     Cross-sectional area when the diameter of a semicircle is $d(x)$?

     Cevap

     The radius is $d(x)/2$, so

     $$A(x)=\frac12\pi\left(\frac{d(x)}2\right)^2=\frac{\pi}{8}[d(x)]^2.$$

268. Kart 268

     Soru

     Why must an area integral be split where curves intersect?

     Cevap

     The top/bottom or right/left relationship may switch there. Splitting keeps each integrand nonnegative and prevents signed cancellation.

269. Kart 269

     Soru

     How can a velocity table approximate displacement?

     Cevap

     Use a left, right, midpoint, or trapezoidal sum for $\int v(t)\,dt$. Each term is velocity times a time width.

270. Kart 270

     Soru

     Washer-method volume formula?

     Cevap

     For outer radius $R(x)$ and inner radius $r(x)$,

     $$V=\pi\int_a^b ([R(x)]^2-[r(x)]^2)\,dx.$$

271. Kart 271

     Soru

     How do you recover position from velocity and an initial position?

     Cevap

     If $s(a)$ is known,

     $$s(t)=s(a)+\int_a^t v(u)\,du.$$

272. Kart 272

     Soru

     How do you choose between vertical and horizontal area slices?

     Cevap

     Choose the direction that describes the region with fewer pieces. Vertical slices use top minus bottom; horizontal slices use right minus left.

273. Kart 273

     Soru

     Cross-sectional area of an equilateral triangle with side $s(x)$?

     Cevap

     $$A(x)=\frac{\sqrt3}{4}[s(x)]^2.$$

274. Kart 274

     Soru

     How can a table approximate the average value of $f$ on $[a,b]$?

     Cevap

     First approximate $\int_a^b f(x)\,dx$ with an appropriate Riemann or trapezoidal sum, then divide by $b-a$.

275. Kart 275

     Soru

     Single expression for area between two curves?

     Cevap

     When the functions are integrable,

     $$A=\int_a^b |f(x)-g(x)|\,dx.$$

     For hand evaluation, split where their order changes.

276. Kart 276

     Soru

     How is a rotation radius measured from a vertical axis $x=k$?

     Cevap

     As horizontal distance: $|x-k|$. With $dy$ slices, write the relevant boundaries as $x$-functions of $y$.

277. Kart 277

     Soru

     How can a rate table approximate total change with unequal time gaps?

     Cevap

     Multiply a representative rate on each interval by that interval's actual width, then sum. Do not assume a common $\Delta t$ when the table is uneven.

278. Kart 278

     Soru

     When should a volume integral use $dy$?

     Cevap

     When slices perpendicular to the $y$-axis make the cross-sectional area easiest to express as $A(y)$. Then use $V=\int_c^d A(y)\,dy$.

279. Kart 279

     Soru

     What signals that a washer, not a disc, is needed?

     Cevap

     The rotated region leaves a hole around the axis. Subtract the inner circular area from the outer circular area.

280. Kart 280

     Soru

     What base length is used for cross sections over a planar region?

     Cevap

     Use the distance across the base region within each slice: top minus bottom for vertical slices, or right minus left for horizontal slices.

281. Kart 281

     Soru

     Why must total distance split at velocity sign changes?

     Cevap

     Distance accumulates speed $|v|$, not signed velocity. A single integral of $v$ would cancel motion in opposite directions.

282. Kart 282

     Soru

     When does an accumulated quantity reach a local maximum?

     Cevap

     When its net rate changes from positive to negative. A zero rate alone is only a candidate; check the sign change.

283. Kart 283

     Soru

     What distinguishes area from a definite integral?

     Cevap

     Area is nonnegative. A definite integral is signed and may be zero or negative because regions below the axis subtract.

284. Kart 284

     Soru

     How do position, velocity, and acceleration graphs correspond?

     Cevap

     Velocity is the slope of position, and acceleration is the slope of velocity. Conversely, signed area under velocity gives position change, and signed area under acceleration gives velocity change.

285. Kart 285

     Soru

     How do you interpret $\int_a^b [r(t)-c(t)]\,dt$ in context?

     Cevap

     As net change: total amount added by $r$ minus total amount removed by $c$ over the interval. State the resulting quantity and units.

286. Kart 286

     Soru

     What units does a volume integral $\int A(x)\,dx$ have?

     Cevap

     Cubic units. Cross-sectional area contributes square units and slice thickness contributes one more length unit.

287. Kart 287

     Soru

     How can a graph of a rate reveal the largest accumulated value?

     Cevap

     Compare accumulation at endpoints and at times where the rate is zero or undefined. Compute signed areas to track the quantity's changes from its initial value.

288. Kart 288

     Soru

     Why should a contextual integral answer include a sentence?

     Cevap

     The number alone does not say whether it represents displacement, distance, area, volume, or net change. Name the quantity, interval, and units.